Problem 57 - University Calculus: Early Transcendentals

Question

In Exercises \(51-70,\) find \(d y / d t\). $$y=e^{\cos ^{2}(\pi t-1)}$$

Step-by-Step Solution

Answer

The derivative is \(-\pi e^{\cos^2(\pi t - 1)} \sin(2(\pi t - 1))\).

Step 1: Identify the Chain Rule Components

The function you need to derive is a composition of functions, meaning we should use the chain rule. The outer function is the exponential function: \(e^u\). The inner function is \(u = \cos^2(\pi t - 1)\). Recognizing these components is crucial for applying the chain rule.

Step 2: Differentiate the Outer Function

Apply the chain rule starting with the derivative of the outer function, which is \(e^u\). The derivative of \(e^u\) with respect to \(u\) is \(e^u\). So, when differentiating \(y = e^{u}\) with respect to \(t\), we get \(e^{u} \cdot \frac{du}{dt}\).

Step 3: Differentiate the Inner Function

Now, differentiate the inner function \(u = \cos^2(\pi t - 1)\). Use the chain rule again: recognize \(u = v^2\) where \(v = \cos(\pi t - 1)\). Differentiate \(u\) with respect to \(v\) to get \(2v\), and \(v\) with respect to \(t\) to get \(-\pi \sin(\pi t - 1)\). So, \(\frac{du}{dt} = 2 \cos(\pi t - 1) \cdot (-\pi \sin(\pi t - 1))\).

Step 4: Combine the Derivatives

Substitute \(u = \cos^2(\pi t - 1)\) and \(\frac{du}{dt} = -2\pi \cos(\pi t - 1) \sin(\pi t - 1)\) into the derivative found in Step 2: \(dy/dt = e^{\cos^2(\pi t - 1)} \cdot (-2\pi \cos(\pi t - 1) \sin(\pi t - 1))\).

Step 5: Simplify the Derivative

Simplify the expression for the derivative: \(dy/dt = -2\pi e^{\cos^2(\pi t - 1)} \cos(\pi t - 1) \sin(\pi t - 1)\). This expression can also be written using the double-angle identity, \(\sin(2\theta) = 2 \sin\theta \cos\theta\), giving: \(dy/dt = -\pi e^{\cos^2(\pi t - 1)} \sin(2(\pi t - 1))\).

Key Concepts

Exponential Functions

Exponential functions are a fundamental part of calculus and appear frequently in differential equations. The basic form of an exponential function is written as \( y = a^x \), where \( a \) is a constant and \( x \) is the exponent. These functions are unique because of their rapid growth rate.
One of the most common exponential functions is the natural exponential function, denoted as \( e^x \), where \( e \) is Euler's number, approximately equal to 2.718. In calculus, the derivative of the exponential function \( e^x \) is quite remarkable because it is the only function whose derivative is itself. This means that if \( y = e^x \), then \( \frac{dy}{dx} = e^x \). This property makes handling calculations involving exponential functions smoother.
Exponential functions are used in various applications like modeling population growth, radioactive decay, and interest calculations. Understanding how these functions work and how they change is key when applying concepts like the chain rule in calculus.

Trigonometric Functions

Trigonometric functions are core elements of calculus, especially in problems involving waves or circular motion. The primary trigonometric functions are sine, cosine, and tangent. In the context of differentiation, these functions have specific rules: the derivative of \( \sin(x) \) is \( \cos(x) \) and the derivative of \( \cos(x) \) is \( -\sin(x) \).
When you have a trigonometric function involved in a composite function, as in the exercise where \( u = \cos^2(\pi t - 1) \), the chain rule is employed. Differentiating \( \cos^2(x) \) requires using both the power rule and the chain rule. First, apply the power rule to get \( 2\cos(x) \). Then, apply the derivative of \( \cos(x) \), which is \( -\sin(x) \). Combining these gives you \( 2 \cos(x) \cdot (-\sin(x)) \).
These properties and rules simplify working with trigonometric functions, whether they are part of a polynomial, exponential like in the original exercise, or any other form.

Derivative Simplification

Simplifying derivatives involves reducing a function's derivative into its simplest form. This not only makes the result more elegant but also easier to interpret.
When simplifying derivatives, use mathematical identities to combine and reduce expressions. For instance, in the given exercise, after applying the chain rule to obtain the derivative, a simplification is performed using the trigonometric identity \( \sin(2\theta) = 2 \sin\theta \cos\theta \). This is used to transform the derivative from \( -2\pi e^{\cos^2(\pi t - 1)} \cos(\pi t - 1) \sin(\pi t - 1) \) into \( -\pi e^{\cos^2(\pi t - 1)} \sin(2(\pi t - 1)) \).
Pulling out constants, combining like terms, or using trigonometric identities are common methods to achieve simplification. This process helps in understanding the behavior of the derivative and its implications in the problem at hand.